Math Answers

) For a scalar field ( , , ) ,
n n n
φ x y z = x + y + z show that (∇φ) r = nφ
r r
. , where n is a non-zero
real constant.

Calculate the volume of the tetrahedron whose vertices are the points A = (3, 2, 1),
B = (1, 2, 4), C = (4, 0, 3) and D = (1, 1, 7).

At a party, you and your friend are both eying the last slice of pizza. To settle the matter, you agree on the following dice game: each of you is going to roll a die; if the highest number rolled by either one of you is a 1, 2, 3 or 4, then Player 1 wins. If the highest number is a 5 or a 6, then Player 2 wins. Assuming that you really want that last slice of pizza would you rather be Player 1 or Player 2 to maximize your chance of winning? Explain your choice.

A survey of 500 construction companies showed that 310 of them advertised in magazine A, 304 advertised in magazine B, and 220 advertised in both.

What is the probability that a construction company advertised in at least one of the two magazines?

1 Evaluate the limit
limh→02(−3+h)2−18h
limh→02(−3+h)2−18h
12
8
14
6
2 Evaluate the limit
limt→4t−(√3+4)4−t
limt→4t−(3+4)4−t

−38
−38

−58
−58

−18
−18

34
34

1 Find the value of
f(x)=x3+x2−5x+1
, when x=2
6
4
3
1
2 Evaluate the value of
f(x−2)
, if
f(x)=x2+5x+5

x2+x+4

x2−x−3

x2+2x+4

x2+x−5

6 Find the integral with respect to x
∫cosxsinxdx

sin2x2+c

sin2x+c

cos2x2+c

sinx
7 Evaluate
∫2−1y2+y−2dy
∫−12y2+y−2dy

716

316

1716

516
8 Integrate with respect to x :
∫2−1x2(x3+4)2dx
12

12
6

512
9 Integrate with respect to x :
∫3−1x7+x2−−−−−√dx
∫−13x7+x2dx

22√
22

42√
42

4−22√
4−22

2√
2
10 Integrate with respect to x :
∫41x+1x√dx

203

20

320

-20

1 Integrate
∫(x3+3x2+2x+4)

x44+x3+x2+4x+c

x42−x3+x2+4x+c

3x44+2x3+x+c

6x4−3x2+c
2 Evaluate
∫(3x−2)6dx

(3x+2)72+c

(3x+2)721+c

(3x−2)721+c

3(3x−2)72+c
3 Evaluate
∫cos(6x+4)dx

sin(6x+4)6+c

cos(6x+4)6+c

tan(6x+4)6+c

sec(6x+4)6+c
4 Evaluate
∫3ex+5cos(x)−10sec2(x)dx

3e2+5sinx−10tanx+c

3ex+cosx−10tanx+c

3ex+5sinx−10secx+c

2ex−x−10tanx+c
5 Evalute
∫x2(3−10x3)dx

1150(3−10x3)5)+c

110(1−10x2)5)+c

115(3−20x3)5)+c

1100(3−2x3)5)+c

7 Determine
∫x2+1(x+2)3

ln(x+2)+4x+2−52(x+3)2+c

ln(x+2)−4x+2−52(x+3)2+c

−ln(x+2)−4x+2−52(x+3)2+c

ln(x−2)+4x−2−52(x+3)2+c

8 Find the volume of a sphere generated by a semicircle
y=(√r2−x)
revolving around the x-axis

−π−r32

4πr32

πr34

4πr33

9 Evaluate
∫x2e3x

e3x3(x2−2x3+29)+c

−e3x3(x2+2x3−29)+c

e2x3(x3−x4+29)+c

ex3(x2+2x3−25)+c
10 Given f(x) =
(7x4−5x3)
, evaluate
df(x)dx

7x4−5x3

2x3−15x2

28x2−15x2

28x3−15x2